The answer to the question as you have asked it is: "No it is not possible"
Moreover dropbox relies on extended attributes (X-attrs) so it needs one of the modern filesystems like ext4 or NTFS. I am certain you cannot bypass that check. And you should not as you risk to loose functionality.
What you can do to solve your problem is convert your FAT to NTFS. This can be done (theoretically) without loss of data, though backup is still recommended:
Find the letter of drive you want to convert. Open File Explorer, and in the left pane, look for the drive under "This PC" or "Computer."
Press Win + R to open the Run dialog box.
Enter cmd to open the command prompt window.
At the command line prompt, type
convert drive: /fs:ntfs.
For example, if the letter of the drive you want to convert is
e, you would type
convert e: /fs:ntfs
Following an update from OP, the original filesystem is exFAT. In this case it is not possible to convert, the only option is to backup and re-format. Good luck
Following yet another update from OP, it turns out that the drive is not actually SSD as initally stated but rather an SD card... In this case best advice is to avoid keeping your dropbox folder on SD card and move it to more robust storage (HDD or SDD). While it is possible to have NTFS filesystem on SD card and it will work, it will not perform very well and it will quickly wear out flash memory of the card.
Worth taking a look at:
Some additional reading: